We study the lifting of representations between the N-fold metaplectic covers of SL(n, F) where n|N and PGL(n, F), F a local field, and obtain a formula relating irreducible characters of PGL(n, F) and SL~(n, F) (the covers of SL(n, F)). This is achieved by generalizing the approach of Adams [1].

In the second part of the thesis we study the lifting of functions between SL~(n, F) and PGL(n, F). Using orbital integrals we obtain the formula for the lifting of characters as a dual to the lifting of functions. This is based on the methods of Flicker and Kazhdan [7].

Finally we use our methods of lifting of orbital integrals to provide alternate proof of a well-known fact about p-adic fields (under certain restrictions). We show that for a Galois extension E/F, F*/N(E*) isomorphic Gal(E/F)ab where Gal(E/F) is the Galois group of E over F and Gal(E/F)ab denotes its abelianization and N: E* to F* is the norm map.